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Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

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Page 1: Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Solving Inequalities, Compound Inequalities and Absolute Value Inequalities

Sec 1.5 &1.6

pg. 33 - 43

Page 2: Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Hallford © 2007 Glencoe © 2003

Objectives

The learner will be able to (TLWBAT): solve inequalities solve real-world problems with

inequalities solve compound inequalities solve absolute value inequalities

Page 3: Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Hallford © 2007 Glencoe © 2003

Properties

Trichotomy Property For any two real numbers a and b,

a < b a = b a > b

Page 4: Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Hallford © 2007 Glencoe © 2003

Properties of Inequalities

For any real number a, b, and c Addition Prop.

If a>b, then a + c > b + c If a<b, then a + c < b + c

Subtraction Prop. If a>b, then a – c > b – c If a<b, then a – c < b – c

See pg. 33

Page 5: Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Hallford © 2007 Glencoe © 2003

Work this problem

x – 7 > 2x + 2 -x -x -7 > x + 2 -2 -2 -9 > x Now we need to graph our answer on

a number line.

Page 6: Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Hallford © 2007 Glencoe © 2003

Graphing on a Number Line

Let’s look at our previous answer, -9 > x

-9 0

What does this answer mean?

Are my “x’s” here?Or here?

-9 is GREATER than x, so my “x’s” that make senseare -10, -10.1, -11, etc. Anything smaller than -9

For < or > you use an opening dot or point -

For < or > you use a closed dot or point -

Page 7: Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Hallford © 2007 Glencoe © 2003

Properties of Inequalities

Multiplication Prop. For any real numbers

a, b, and c If c is positive If a>b, then ac>bc If a<b, then ac<bc

If c is negative If a>b, then ac<bc If a<b, then ac>bc

Division Prop. For any real numbers

a, b, and c If c is positive If a>b, then a/c>b/c

If a<b, then a/c<b/c

If c is negative If a>b, then a/c<b/c

If a<b, then a/c>b/c

Page 8: Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Hallford © 2007 Glencoe © 2003

Work this problem

3x – 7 < 7x + 13

-4x – 7 < 13

7 7

-4x < 20

-4 -4

x > -5

Now let’s graph

-7x -7x

0-5

You can also use set builder notation toexpress your answer.

{x | x > -5}

This is read as the set of all x such that x is greater than or equal to -5

Page 9: Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Hallford © 2007 Glencoe © 2003

Work this problem

3(a +4) – 2(3a +4) ≤ 4a - 1

3a + 12 – 6a – 8 ≤ 4a – 1

-3a + 4 ≤ 4a – 1

4 ≤ 7a - 1

5 ≤ 7a5/7 ≤ a

Now graph andexpress in setbuilder notation.

0

{a | a ≥ 5/7}

You can also use interval notation. Interval notation uses ( & ) for < or > and [ & ] for ≤ & ≥. We also use -∞ (negative infinity) & +∞ (positive infinity)

Interval notation for this problem would be [5/7, +∞)

Page 10: Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Hallford © 2007 Glencoe © 2003

Compound Inequalities

A compound inequality consists of two inequalities joined by the word “and” or the word “or”.

You must solve both inequalities and then graph. The final graph of the “and” inequality is the

intersection of both individual solution graphs. The final graph of the “or” inequality in the union

of both individual solution graphs

Page 11: Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Hallford © 2007 Glencoe © 2003

Let’s solve an “and” compound inequality

7 < 2x - 1 < 15

Method 1 – Divide into two problems

7 < 2x – 18 < 2x4 < x

Method 2 – Work together

2x – 1 < 152x < 16x < 8

7 < 2x – 1 < 158 < 2x < 164 < x < 8

Whateveryou do toone sidedo to the other!

Now Graph

4 < x

x < 8

0

0

0

4

4

4

8

8

8

{ x | 4 ≤ x < 8}

“Means 7 < 2x -1 and 2x – 1< 15”

4 < x < 8

Page 12: Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Hallford © 2007 Glencoe © 2003

Let’s work about the same problem as an “or” inequality

7 < 2x -1 or 2x – 1 < 15

We know the answer is 4 < x and x < 8, but thistime the answer graph is different. It is the UNIONof the two graphs.

4 < x

x < 8

0

0

0

4

4

8

8

The solution set is all real numbers .

4 < x or x < 8

Page 13: Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Hallford © 2007 Glencoe © 2003

Work this problem

2x + 7 < -1 or 3x + 7 > 10

2x + 7 < -1 3x + 7 > 10

2x < -8x < -4

3x > 3x > 1

0-4

0

or

1

0-4 1

Page 14: Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Hallford © 2007 Glencoe © 2003

Absolute Value Inequalities

|a| < b, where b > 0 work as an “and” problem -b < a < b

|a| > b, where b > 0 work as an “or” problem a > b or a < -b

Page 15: Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Hallford © 2007 Glencoe © 2003

Work these problems

|x – 1| < 3 -3 < x -1 < 3 -2 < x < 4

|x -1 | > 3 x -1 > 3 or x -1 < -3 x > 4 or x < -2

0 0

Page 16: Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Hallford © 2007 Glencoe © 2003

Absolute Value Inequalities

|a| < b, where b < 0 if b is less than zero, it is negative, so

there is no solution |a| > b, where b < 0

if b is less than zero, it is negative, so every real number is a solution or all reals.

Page 17: Solving Inequalities, Compound Inequalities and Absolute Value Inequalities Sec 1.5 &1.6 pg. 33 - 43

Hallford © 2007 Glencoe © 2003

Work these problems

|2x – 7| < -5

There is no solutionfor the above sincethe absolute value cannot be less than zero.

|3x – 1| + 9 > 2

|3x – 1| > -7

Any value of “x” willmake this statementtrue, since the absolute value is alwaysgreater than a negativenumber

All Real numbers